[1003.0481]
The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations

Authors:

Cyril Pitrou, Jean-Philippe Uzan, Francis Bernardeau

Abstract:

This article presents the first computation of the complete bispectrum of the
cosmic microwave background temperature anisotropies arising from the evolution
of all cosmic fluids up to second order, including neutrinos. Gravitational
couplings, electron density fluctuations and the second order Boltzmann
equation are fully taken into account. Comparison to limiting cases that
appeared previously in the literature are provided. These are regimes for which
analytical insights can be given. The final results are expressed in terms of
equivalent fNL for different configurations. It is found that for moments up to
lmax=2000, the signal generated by non-linear effects is equivalent to fNL~5
for both local-type and equilateral-type primordial non-Gaussianity.

This is a tour de force paper in which authors cobble together a second-order perturbation code using Mathematica (publicly available here: http://icg.port.ac.uk/~pitrouc/cmbquick.htm). This allows them to calculate, for the first time, what is the CMB bispectrum induced by 2nd order perturbation theory. I have read the paper only cursorily and there are a few interesting things to note:

a) The main result is that the effective fnl, measured by future expts will be ~5. I think people have arrived to this number before, but this is the first time it has been shown explicitly.

b) 2nd perturbation theory induces spectral distortions and so results depend on what you define to be your temperature. They use something they call "bolometric temperature", defined as "temperature of the black body which carries the same energy density as the observed distribution". But it is a subtle issue. It would be interesting if one would measure deviations from black body as a function of position and then do the power-spectrum of that!

c) They use a "flat sky" approximation and claim to be a good approx for l>10... I am somewhat confused about that.

d) There is no discussion of the second order corrections to the power spectrum, at least not from my quick reading of the paper. They do make seem to make rather rough approximations when calculating power spectrum, like flat sky and thin shell, but they never show a plot of C_\ell s. The point is that corrections are probably of the order of 10^{-5}, but it'd be still interesting to show explicitly that they're negligible.

It might certainly be interesting to see changes to the power spectrum from around LSS, but in general calculating C_l requires at least 3-rd order perturbation theory since 2nd order squared is the same order as 3rd order times first order: in fact the lensing contributions from 2x2 terms are large and mostly cancel with 3x1 terms, so a C_l ignoring 3rd order terms should differ significantly from the linear-theory result.

I recall there was some controversy about the effect of perturbed recombination, e.g. 0903.0871 and 0812.3658, which this paper appears not to comment on on the grounds that the relevant terms are not gauge invariant. However I would have thought with a suitably clear definition of which terms one is calculating it should be possible to make a meaningful comparison.